To be honest, it’s not the dog that’s the problem, it’s your dh and dd. Dogs do bring a lot of extra work and mess, but it sounds from your op that they aren’t doing anything to help, dog care or otherwise.
Having dogs does mean you will, barring regular cleaners coming in and you topping up daily as well, most likely have to drop your standards in terms of mess. There’s always going to be mud and leaves tracked in, fur everywhere toys shredded etc, but if you are a three person household and all wanted the dog it should be a shared responsibility to care for and clean up after them.
I do understand, I have one older teen with a chronic illness and one who is neurodiverse with mh issues that mean they are unable to help much, plus a younger dc who is just becoming a teen and developing the ‘Kevin the Teenager’ attitude, so every request is a battle. BUT, everyone can and should do something.
In my experience the only way to address it is to make things uncomfortable for them, as a pp said, no lifts, don’t run around after them, if they don’t have what they need it’s on them. I will admit to being hopeless at this myself, but I have had to learn to stop myself constantly being there for everyone except myself. It goes back to that age old ‘saying put your own oxygen mask on first’. If you go under because they haven’t pulled their weight then things will be a lot worse for everyone.
Given what you went through last year, have you sat down and actually had a calm, but frank, discussion about it all with them? Explain how you are feeling and where you think they could help, then agree specific jobs they can take over. Carve out some time for yourself and tell them at this time on x day I won’t be available.
In addition, if your husband is always working and simply not around to help, maybe a dog walker - even just one day a week - to take some of the stress off you and give you a break. If you walking with your pup would actually benefit your mental health, could you maybe prioritise that and farm out your washing and ironing to a service and/or pay a cleaner to keep on top of the basics?